Monstrosity is defined as giving the creature the ability to become monstrous. Monstrosity is an activated ability you can activate any time you can pay for it. When it resolves, it checks whether the creature is already monstrous. If it's not, the ability puts a number of +1/+1 counters on the creature and causes the creature to become monstrous.   Monstrosity skipped Born of the Gods, but reappeared in Journey into Nyx.
Monstrous is more like a characteristic of a card. Many creatures with a monstrosity ability have a triggered ability that triggers when the creature becomes monstrous. There are also some cards with monstrosity that gain abilities as long as they are monstrous.
- Although it is not a verb, it is considered a keyword action in the Comprehensive Rules
- Once a creature becomes monstrous, it can't become monstrous again. If the creature is already monstrous when the monstrosity ability resolves, nothing happens.
- Monstrous isn't an ability that a creature has. It's just something true about that creature. If the creature stops being a creature or loses its abilities, it will continue to be monstrous.
- An ability that triggers when a creature becomes monstrous won't trigger if that creature isn't on the battlefield when its monstrosity ability resolves.